How to find $P(X_1 + X_2 + X_3 + X_4 \geq 3)$ for uniformly distributed independent random variables $X_1$, $X_2$, $X_3$, $X_4\sim U(0,1)$?
It follows from independence that their cumulative density function is 1, but I'm struggling with integration space.
Once we get to the step proposed by @Did, we can obtain the solution easily using geometric probability. Our probability here would be the hyper-volume covered by $$x_1+x_2+x_3+x_4 \leq 1 \text{ and } 0\leq x_i \leq 1$$ is exactly the hyper-volume of an $4$-dimensional simplex, which is $\dfrac{1}{4!} = \dfrac{1}{24}$.